Binomial theorem
The Binomial Theorem
The Binomial Theorem is a topic in Polynomial Functions in the Common Core State Standards. It is aligned to Standard A-APR.5, which requires students to use the Binomial Theorem to expand a power of a binomial, with coefficients from Pascal's triangle.
The Binomial Theorem expands \((a+b)^n\) using the binomial coefficients — the rows of Pascal's triangle.
Theory
The Binomial Theorem expands a power of a binomial:
where \(\binom{n}{k}\) is a binomial coefficient — the entries in row \(n\) of Pascal's triangle.
The expansion:
How to expand
- Read the coefficients from row \(n\) of Pascal's triangle.
- Write descending powers of \(a\) and ascending powers of \(b\).
- Multiply in each coefficient.
- Simplify any numerical parts.
Use row 3 coefficients \(1,3,3,1\).
| \((a+b)^3\) | \(=\) | \(a^3+3a^2b+3ab^2+b^3\) |
Substitute \(a=x,\ b=2\).
| \(x^3+3x^2(2)+3x(4)+8\) | ||
| \(=\) | \(x^3+6x^2+12x+8\) |
The term with \(b^2\) uses \(\binom{4}{2}=6\).
| \(\binom{4}{2}x^2 3^2\) | \(=\) | \(6\cdot 9\,x^2\) |
| \(=\) | \(54x^2\) |
Read row 4 of Pascal's triangle.
| \(1,\ 4,\ 6,\ 4,\ 1\) |
Common pitfalls
Frequently asked questions
What is the Binomial Theorem?
A formula expanding \((a+b)^n\) using binomial coefficients.
Where do the coefficients come from?
Row \(n\) of Pascal's triangle, or \(\binom{n}{k}\).
How do the exponents behave?
Powers of \(a\) decrease while powers of \(b\) increase; each term's exponents add to \(n\).
How many terms are in the expansion?
\(n+1\) terms for \((a+b)^n\).